LOGISTIC Procedure
The SAS LOGISTIC procedure can be also used for a probit regression. To fit a probit regression use the LINK=NORMIT (or PROBIT) option:
proc logistic; model y=x1 x2 / link=normit; run;
Using the data in Example 1, you can use:
proc logistic data=ingot; model s=t / link=normit; run;
You will have the SAS output:
Sample Program: Probit Regression
The LOGISTIC Procedure
Data Set: WORK.INGOT
Response Variable: S
Response Levels: 2
Number of Observations: 387
Link Function: Normit
Response Profile
Ordered
Value S Count
1 0 12
2 1 375
Model Fitting Information and Testing Global Null Hypothesis BETA=0
Intercept
Intercept and
Criterion Only Covariates Chi-Square for Covariates
AIC 108.988 99.018 .
SC 112.947 106.934 .
-2 LOG L 106.988 95.018 11.971 with 1 DF (p=0.0005)
Score . . 15.100 with 1 DF (p=0.0001)
Analysis of Maximum Likelihood Estimates
Parameter Standard Wald Pr > Standardized
Variable DF Estimate Error Chi-Square Chi-Square Estimate
INTERCPT 1 -2.8004 0.3284 72.7050 0.0001 .
T 1 0.0391 0.0113 11.9525 0.0005 0.388259
Association of Predicted Probabilities and Observed Responses
Concordant = 59.2% Somers' D = 0.499
Discordant = 9.4% Gamma = 0.727
Tied = 31.4% Tau-a = 0.030
(4500 pairs) c = 0.749
PROBIT Procedure
You can use the PROC PROBIT to fit a probit model. The basic syntax you can use is:
proc probit; class y; model y=x1 x2; run;
or
proc probit; model r/n=x1 x2; run;
or
proc probit; class x2; model r/n=x1 x2; run;
depending on the nature of the data set.
Using the data in Example 1, you can use:
proc probit data=ingot; class s; model s=t; run;
You will have the following SAS output:
Sample Program: Probit Regression
Probit Procedure
Class Level Information
Class Levels Values
S 2 0 1
Number of observations used = 387
Probit Procedure
Data Set =WORK.INGOT
Dependent Variable=S
Weighted Frequency Counts for the Ordered Response Categories
Level Count
0 12
1 375
Log Likelihood for NORMAL -47.5087804
Probit Procedure
Variable DF Estimate Std Err ChiSquare Pr>Chi Label/Value
INTERCPT 1 -2.8003508 0.331621 71.30839 0.0001 Intercept
T 1 0.0390757 0.011425 11.69807 0.0006
Probit Model in Terms of Tolerance Distribution
MU SIGMA
71.66476 25.59135
Estimated Covariance Matrix for Tolerance Parameters
MU SIGMA
MU 186.336614 98.799500
SIGMA 98.799500 55.985053
Using the data in Example 7, if you use:
proc probit data=drug; class drug; model r/n=x drug; run;
you will have the result:
Sample Program: Probit Regression
Probit Procedure
Class Level Information
Class Levels Values
DRUG 5 A B C D E
Number of observations used = 18
Probit Procedure
Data Set =WORK.DRUG
Dependent Variable=R
Dependent Variable=N
Number of Observations= 18
Number of Events = 99 Number of Trials = 237
Log Likelihood for NORMAL -114.6516555
Probit Procedure
Variable DF Estimate Std Err ChiSquare Pr>Chi Label/Value
INTERCPT 1 0.19031335 0.24926 0.582954 0.4452 Intercept
X 1 1.15885442 0.438333 6.989539 0.0082
DRUG 4 64.33502 0.0001
1 -1.7087998 0.331686 26.5416 0.0001 A
1 -1.2286831 0.239099 26.40741 0.0001 B
1 -2.2309708 0.343196 42.2574 0.0001 C
1 -0.5079719 0.291889 3.028612 0.0818 D
0 0 0 . . E
SAS PROC PROBIT models the probability of Y=0 or of Y's lower sorted value by default. This default can be altered by using the ORDER option in the PROC PROBIT statement. For example,
proc probit order=freq;
specifies the sorting order for the levels of the classification variables (specified in the CLASS statement) in a descending frequency count; levels with the most observations come first in the order.
You may need to model the probability of the value with the higher count. In Example 12, Y=1 has the count 375 and Y=0 has 12. If you use:
proc probit order=freq data=ingot; class s; model s=t; run;
you will have the following output:
Sample Program: Probit Regression
Probit Procedure
Class Level Information
Class Levels Values
S 2 1 0
Number of observations used = 387
Probit Procedure
Data Set =WORK.INGOT
Dependent Variable=S
Weighted Frequency Counts for the Ordered Response Categories
Level Count
1 375
0 12
Log Likelihood for NORMAL -47.5087804
Probit Procedure
Variable DF Estimate Std Err ChiSquare Pr>Chi Label/Value
INTERCPT 1 2.80035085 0.331621 71.30839 0.0001 Intercept
T 1 -0.0390757 0.011425 11.69807 0.0006
Probit Model in Terms of Tolerance Distribution
MU SIGMA
71.66476 25.59135
Estimated Covariance Matrix for Tolerance Parameters
MU SIGMA
MU 186.336614 98.799500
SIGMA 98.799500 55.985053
Sometimes you will need to know the predicted probability values. For example, if you need to know the probability of having an ingot not ready for rolling (Y=0) at T=7 from Example 12, you can compute the probability using the formula:
from the standard normal probability distribution table. You can obtain this kind of computation using the OUTPUT statement and the PRINT procedure:
proc probit; model r/n=x1 x2; output out=filename prob=varname; run; proc print data=filename; run;
where filename is the output data set name and varname is the variable name for predicted probabilities. The SAS output will show all the predicted probabilities for all observation points.
proc probit data=ingot2; model r/n=t; output out=prob2 prob=phat; run; proc print data=prob2; run;
you will have the following additional result:
Sample Program: Probit Regression
OBS T S N PHAT
1 7 0 55 0.00576
2 14 2 157 0.01212
3 27 7 159 0.04047
4 51 3 16 0.20969
GENMOD Procedure
Probit regression can be modeled as a class of generalized linear models in which the response probability function is binomial and the link function is probit. Therefore you can use the PROC GENMOD to fit a probit model:
proc genmod; model r/n=x1 x2 / dist=binomial link=probit; run;
Using the data as in Example 2, you may use:
proc genmod data=ingot2; model r/n=t / dist=binomial link=probit; run;
You will have the following SAS output:
Sample Program: Probit Regression
The GENMOD Procedure
Model Information
Description Value
Data Set WORK.INGOT2
Distribution BINOMIAL
Link Function PROBIT
Dependent Variable R
Dependent Variable N
Observations Used 4
Number Of Events 12
Number Of Trials 387
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 2 0.7392 0.3696
Scaled Deviance 2 0.7392 0.3696
Pearson Chi-Square 2 0.4228 0.2114
Scaled Pearson X2 2 0.4228 0.2114
Log Likelihood . -47.5088 .
Analysis Of Parameter Estimates
Parameter DF Estimate Std Err ChiSquare Pr>Chi
INTERCEPT 1 -2.8004 0.3316 71.3084 0.0001
T 1 0.0391 0.0114 11.6981 0.0006
SCALE 0 1.0000 0.0000 . .
NOTE: The scale parameter was held fixed.
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